Sequences

Correct answer: B. \((0,3,8)\)

Explanation: For the first three terms, substitute \(n=1,2,3\). This gives \(a_1=1^2-1=0\), \(a_2=2^2-1=3\), and \(a_3=3^2-1=8\). Hence, the correct sequence is \((0,3,8)\). Option \((1,4,9)\) uses only \(n^2\) and does not subtract 1. Exam tip: to find initial terms from an nth-term rule, substitute 1, 2, and 3 in order and check each calculation.

Correct answer: C. (2, 4, 8, 16)

Explanation: For the first term, take \(n=1\). Thus, \(a_1=2^1=2\), \(a_2=2^2=4\), \(a_3=2^3=8\), and \(a_4=2^4=16\). Hence, the correct sequence is \((2, 4, 8, 16)\). Option A starts with \(2^0\), but the first term conventionally corresponds to \(n=1\). Exam tip: To list the first four terms, substitute \(n=1,2,3,4\) in the given rule.

Correct answer: D. 28

Explanation: This is an arithmetic sequence because 4 is added to each successive term. The terms are 8, 12, 16, 20, 24, 28; therefore, the 6th term is 28. Although 24 is a close distractor, it is the 5th term. Exam tip: write the terms in order and count their positions carefully.

Correct answer: A. 60

Explanation: In this sequence, each successive term is 6 less than the previous one: 90, 84, 78, 72, 66, 60. Therefore, the 6th term is 60. The number 66 is the 5th term, so it is a close but incorrect option. Exam tip: Count the first listed term as the 1st term.

Correct answer: B. 7th

Explanation: Each term of this sequence is a multiple of 5, so its nth term is \(5n\). Putting \(5n=35\) gives \(n=7\); hence, 35 is the 7th term. The 6th term is \(30\), so the closest distractor is incorrect. Exam tip: To find a term’s position, equate the general term to the given value.

Correct answer: C. fifth

Explanation: This is an arithmetic sequence with first term 3 and common difference 4. Its terms are 3, 7, 11, 15, 19; therefore, 19 is the fifth term. The fourth term is 15, so the fourth option is not correct. Exam tip: list the terms in order and count their positions.

Correct answer: D. fifth

Explanation: Each successive term is 4 less than the preceding term: 48, 44, 40, 36, 32. Hence, 32 is the fifth term. The fourth term is 36, so the fourth-term option is incorrect. Exam tip: To find a term’s position, write the sequence with term numbers 1, 2, 3, and so on.

Correct answer: A. (6, 12, 18, 24)

Explanation: In the sequence (6, 12, 18, 24), the first term is 6, the second is 12, and the third is 18. Therefore, option A is correct. In option C, 18 is the second term, not the third. In exams, count terms from the left to identify their positions.

Correct answer: B. (7, 12, 17, 22)

Explanation: In the sequence (7, 12, 17, 22), the first term is 7 and the consecutive differences are 12 - 7 = 5, 17 - 12 = 5, and 22 - 17 = 5. Therefore, option B is correct. In option A, 4 is added each time, not 5. In exams, check both the first term and the difference between consecutive terms.

Correct answer: C. \(64, 56, 48, 40\)

Explanation: In the sequence \(64, 56, 48, 40\), the differences between consecutive terms are \(56-64=-8\), \(48-56=-8\), and \(40-48=-8\). Thus, 8 is subtracted each time. In option D, the terms decrease by 16, so it is not correct. Exam tip: Find the difference between consecutive terms to check a sequence rule quickly.

Correct answer: D. 21

Explanation: The terms of the sequence are 13, 17, 21, 25, \(\ldots\). Here, \(a_3\) denotes the third term, so \(a_3=21\). The number 25 is the fourth term, not the third. Exam tip: the subscript \(n\) in \(a_n\) indicates the position of the term.

Correct answer: A. \(49\)

Explanation: The terms of the sequence are 70, 63, 56, and 49 in order. Therefore, the fourth term, \(a_4\), is \(49\). The number \(56\) is the third term, so it cannot be \(a_4\). Exam tip: To identify \(a_n\), count the terms from the left as 1, 2, 3, and so on.

Correct answer: B. 11

Explanation: In a sequence, \(a_n\) denotes the \(n\)th term. Hence, \(a_2\) is the second term. The given terms are 2, 11, 20, and 29, so \(a_2=11\). The number 20 is the third term, \(a_3\), not the second term. Exam tip: Count terms from the first term when reading sequence notation.

Correct answer: C. 18

Explanation: In a sequence, \(a_1\) denotes the first term. The first term of \(18,24,30,36,\ldots\) is 18, so 18 is correct. Here, 24 is the second term, \(a_2\), not \(a_1\). Exam tip: \(a_1\), \(a_2\), and \(a_3\) represent the first, second, and third terms respectively.

Correct answer: D. 21

Explanation: The differences between consecutive terms are 2, 3, 4, and 5. Since each difference increases by 1, the next difference is 6. Therefore, the next term is \(15+6=21\). Choosing 20 would give a difference of 5, but the required next difference is 6. Exam tip: For such sequences, list consecutive differences first and look for their pattern.

Correct answer: A. 37

Explanation: The differences between consecutive terms are 3, 5, 7, and 9, which are consecutive odd numbers. Therefore, the next difference is 11. Hence, the next term is \(26+11=37\). Choosing 35 would give a difference of only 9 and would break the pattern of increasing odd differences. Exam tip: For such sequences, first write the consecutive differences to identify the pattern.

Correct answer: B. 25

Explanation: The differences between consecutive terms are \(7-5=2\), \(10-7=3\), \(14-10=4\), and \(19-14=5\). Since the difference increases by 1 each time, the next difference is \(6\). Therefore, the next term is \(19+6=25\). Option 24 would result if the next difference remained 5, which does not follow the pattern. Exam tip: Write the consecutive differences first to identify a sequence pattern.

Correct answer: C. 7.5

Explanation: In this sequence, each term is obtained by dividing the previous term by 2: 120 to 60, 60 to 30, and 30 to 15. Therefore, the next term is \(15 \div 2 = 7.5\). Option 8 is not half of 15. Exam tip: Check for a common multiplication or division rule between consecutive terms.

Correct answer: D. 19

Explanation: To find the third term, substitute \(n=3\) in the rule: \(a_3=6\times3+1=18+1=19\). Hence, 19 is the correct answer. The value 18 is only \(6\times3\), so it misses the \(+1\) in the rule. Exam tip: substitute the given value of \(n\) first, then carry out all operations carefully.

Correct answer: A. 12

Explanation: Given \(a_n=3n^2\), substitute \(n=2\): \(a_2=3\times2^2=3\times4=12\). Hence, option A is correct. Option B results from multiplying \(3\times2\) without squaring 2. Exam tip: while finding a term of a sequence, substitute the given value of \(n\) first and then evaluate the exponent.

Correct answer: B. (13, 11, 9)

Explanation: To find the first three terms, substitute n=1, 2, and 3. This gives a_1=15-2(1)=13, a_2=15-2(2)=11, and a_3=15-2(3)=9. Therefore, the sequence begins with (13, 11, 9). The option (15, 13, 11) would result if indexing started at n=0. Exam tip: Unless stated otherwise, use n=1 to find the first term from a_n.

Correct answer: C. 160

Explanation: Each term in the sequence is twice the preceding term: 10, 20, 40, 80. Therefore, the next term is \(80\times 2=160\). The value 120 would result from adding 40 to 80, but the pattern is multiplication by 2, not addition. Exam tip: check the ratio of consecutive terms; here \(20/10=40/20=80/40=2\).

Correct answer: D. 5

Explanation: Write the sequence in pairs: (2, 4), (3, 6), (4, 8). In each pair, the second term is twice the first term, while the first terms 2, 3, 4, ... increase by 1. Therefore, the term after 8 is 5. Option 10 is not the next term; it would come after 5. Exam tip: In such sequences, examine the odd-position and even-position terms separately to identify the pattern.

Correct answer: B. 15

Explanation: The differences between consecutive terms are \(2, 4, 2, 4\). The pattern alternates between \(+2\) and \(+4\), so add \(+2\) after 13: \(13+2=15\). Therefore, 15 is the correct answer. Option 17 would result from adding 4 next, but the next difference in the pattern is 2. Exam tip: For next-term questions, first write the differences between consecutive terms to identify the pattern.

Correct answer: C. 4, 9, 16, 25

Explanation: The governing concept is an arithmetic sequence, whose consecutive terms have one constant difference. Check each option by subtracting every term from the next. In A, the differences are 10−7=3, 13−10=3, and 16−13=3, so the difference is constant. In B, they are 16−20=−4, 12−16=−4, and 8−12=−4. In D, they are 11−5=6, 17−11=6, and 23−17=6. In C, however, the differences are 9−4=5, 16−9=7, and 25−16=9. Since these are unequal, C is not an arithmetic sequence and is correct. Its terms are consecutive squares, but that observation is not required.